Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.

Abstract. Consider a system Ψ of non-constant affine-linear forms ψ1,..., ψt: Z d → Z, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [−N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N → ∞, for the number of integer points n ∈ Z d ∩ K for which the integers ψ1(n),..., ψt(n) are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms ψ1,..., ψt are affinely related; this excludes the important “binary ” cases such as the twin prime or Goldbach conjectures, but does allow one to count “non-degenerate ” configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowers-norm conjecture (GI(s)) and the Möbius and nilsequences conjecture (MN(s)), where s ∈ {1, 2,...} is